On solutions of real analytic equations

Abstract:

Analyticity of $\mathcal {C}^\infty$ solutions $y_i =f_i(x), 1\le i\le m$, of systems of real analytic equations $p_j(x,y)= 0, 1\le j\le l$, is studied. Sufficient conditions for $\mathcal {C}^\infty$ and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by $p_1,p_2,\ldots ,p_l$. In a special case when the $p_i$’s are independent of $x$, we prove that if a $\mathcal {C}^\infty$ solution $h$ satisfies the condition $\det \left ( \frac {\partial p_i}{py_j}\right )(h(x)) \not \equiv 0$, then $h$ is necessarily real analytic.